Modern Stochastics: Theory and Applications

Let $g(s,t)$ be the solution of (2), and the process Z be defined by (5). Then the following assertions hold:

Let $g(s,t)$ be the solution of (2), and let the processes Q and Z be defined by (4) and (5), respectively. The maximum likelihood estimator of ϑ is given by Since $\vartheta >0$, this estimator is asymptotically normal at the usual rate:

The maximum likelihood estimator $\widehat{\vartheta }_{T}$ defined by (9) satisfies the large deviation principle with the good rate function

As it was mentioned in the previous section, in order to establish the large deviation principle for $\widehat{\vartheta }_{T}$ and determine the corresponding good rate function, it is necessary to find the limit and determine the set of $(a,b)\in {\mathbb{R}}^{2}$ for which this limit is finite.

For arbitrary $\varphi \in \mathbb{R}$, consider the Doleans exponential of $(\varphi +\vartheta ){\int _{0}^{t}}Q_{s}dM_{s}$,

Note that $(\frac{1}{\sqrt{\psi (t,t)}}Q_{t})_{t\ge 0}$ is a Gaussian process whose mean and variance functions are bounded on $[0,T]$. Thus, $\varLambda _{\varphi }$ satisfies the conditions of Girsanov’s theorem in accordance with Example 3 of paragraph 2 of Section 6 in [11], and we can apply a usual change of measures and consider the new probability $\mathbb{P}_{\varphi }$ defined by the local density

Observe that, due to (6), $\varLambda _{\varphi }(T)$ can be rewritten in terms of the fundamental semimartingale Consequently, we can rewrite $\mathcal{L}_{T}(a,b)$ as Given an arbitrary real number φ, we can choose $\varphi =a-\vartheta $. Then or, denoting $\mu =-\frac{1}{2}(2b-{\vartheta }^{2}+{(a-\vartheta )}^{2})$, As it was mentioned before, the expectation in (13) can be infinite for some combinations of μ and φ. Our purpose is to determine the set of $(\varphi ,\mu )\in {\mathbb{R}}^{2}$ for which this expectation and limit (12) are finite. According to Girsanov’s theorem, under $\mathbb{P}_{\varphi }$, the process has the same distribution as M under $\mathbb{P}$. Consequently, applying the inverse integral transformation (7) to (14), we get that, under $\mathbb{P}_{\varphi }$, the process $X_{t}-\varphi {\int _{0}^{t}}X_{s}ds$ is a mixed fractional Brownian motion.

Under the new probability measure $\mathbb{P}_{\varphi }$, the process X is a mixed fractional Ornstein–Uhlenbeck process with drift parameter $-\varphi $. Consequently, in order to find the limit of (13) as $T\to \infty $, we can apply Lemma 1, which is presented in Section 5. Thus, we have the equality and convergence (12) holds for $\mu >-\frac{{\varphi }^{2}}{2}$, which gives ${\vartheta }^{2}-2b>0$.

For $x\in \mathbb{R}$, denote the function defined on the set Then, according to (11), the rate function $I(x)$ for the maximum likelihood estimator $\widehat{\vartheta }_{T}$ can be found as Consequently, straightforward calculations of this infimum finish the proof of the theorem.  □

Observe that the rate function $I(x)$ does not depend on the parameter H. Hence, $\widehat{\vartheta }_{T}$ shares the same large deviation principles as those established by Florens-Landais and Pham [8] for a standard Ornstein–Uhlenbeck process and by Bercu, Coutin, and Savy [3] for a fractional Ornstein–Uhlenbeck process (see also [2, 9]).

For a mixed fractional Ornstein–Uhlenbeck process X with drift parameter ϑ, we have the following limit: for all $\mu >-\frac{{\vartheta }^{2}}{2}$.

We shall prove the lemma using an approach similar to that in [6]. Denote $V_{t}={\int _{0}^{t}}\psi (s,s)dZ_{s}$. Then, according to (8), we can rewrite where $W_{t}$ is a Brownian motion. Consequently, we get

The Gaussian vector $\zeta _{t}={(Z_{t},V_{t})}^{T}$ is a solution of the linear system of the Itô stochastic differential equation where Moreover, $\mathcal{K}_{T}(\mu )$ in (15) can be rewritten as where By the Cameron–Martin-type formula from Section 4.1 of [10], where $\varGamma (t)$ is the solution of the equation with $B(t)=b(t){b}^{T}(t)$ and initial condition $\varGamma (0)=0$.

We shall search solution of (16) as the ratio $\varGamma (t)={\varPsi _{1}^{-1}}(t)\varPsi _{2}(t)$, where $\varPsi _{1}(t)$ and $\varPsi _{2}(t)$ are the solutions of the following equation system: with initial conditions $\varPsi _{1}(0)=I$ and $\varPsi _{2}(0)=0$. From the first equation of (17) we get and since $\text{tr }A(t)=2$, it follows that Since $\dot{\varPsi }_{1}(t)=\varPsi _{1}(t)({\varPsi _{1}^{-1}}(t)\dot{\varPsi }_{1}(t))$, by Liouville’s formula we have In order to calculate $\lim _{T\to \infty }\frac{1}{T}\log \det \varPsi _{1}(T)$, define the matrix and note that $A{(t)}^{T}=JA(t)J$, $R(t)=JA(t)$, and $B(t)=A(t)J$. Setting $\widetilde{\varPsi }_{2}(t)=\varPsi _{2}(t)J$, from (17) we obtain the following equation system: with initial conditions $\varPsi _{1}(0)=I$ and $\widetilde{\varPsi }_{2}(0)=0$. When $\frac{{\vartheta }^{2}}{2}+\mu >0$, the coefficient matrix of system (18) has two real eigenvalues $\pm \lambda $ with $\lambda =\sqrt{\frac{{\vartheta }^{2}}{4}+\frac{\mu }{2}}$ and eigenvectors Denote ${a}^{\pm }=\frac{\vartheta }{2}\pm \lambda =\frac{\vartheta }{2}\pm \sqrt{\frac{{\vartheta }^{2}}{4}+\frac{\mu }{2}}$. Diagonalizing system (18), we get where $\varUpsilon _{1}(t)$ and $\varUpsilon _{2}(t)$ are the solutions of the equations with initial conditions $\varUpsilon _{1}(0)=\frac{1}{2\lambda }I$ and $\varUpsilon _{2}(0)=-\frac{1}{2\lambda }I$. Denote the matrix $M(T)={\varUpsilon _{2}^{-1}}(T)\varUpsilon _{1}(T)$, which is the solution of the equation subject to initial condition $M(0)=-I$. Then Applying Liouville’s formula to (19), we get as $T\to \infty $. Thus, in order to prove that limit (15) holds, we should show that Given (20), by Theorem 3 in [13] we have Thus, limit (21) holds, which finishes the proof of the lemma.  □